Lattices Generated by Orbits of Subspaces in $t$-Singular Linear Spaces

Abstract

For non-negative integers $n_1,n_2,\dots ,n_t$, let $G{L}_{n_1,n_2,\dots ,n_t}({\mathbb{F}}_q)$ denote the $t$-singular general linear group of degree $n=n_1+n_2+\cdots +n_t$ over the finite field ${\mathbb{F}}_q^{n_1+n_2+\dots +n_t}$ denote the $(n_1+n_2+\dots +n_t)$-dimensional $t$-singular linear space over the finite $\mathbb{F}$. Let $\mathcal{M}$ be any orbit of subspaces under $G{L}_{n_1,n_2,\dots ,n_t}({\mathbb{F}}_q)$. Denote by $\mathcal{L}$ the set of all intersections of subspaces in $M$. Ordered by ordinary or reverse inclusion, two posets are obtained. This paper discusses their geometricity and computes their characteristic polynomials.